<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K5" target="_self">lim_sup</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:13_1"/><i><font color="Green">E11</font></i>: 
(  <a href="kurato_2.html#K5" target="_self">lim_sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K1" target="_self">Union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="kurato_2.html#D4" target="_self">Def4</a></i>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E4:13_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K1" target="_self">Union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="kurato_2.html#E3:13_1"><i><font color="Green">E11</font></i></a></i>;<br/><a NAME="E5:13_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K1" target="_self">Union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="kurato_2.html#E1:13_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="kurato_2.html#E3:13_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="kurato_2.html#T6" target="_self">Th6</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:13_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><b>thus </b><a NAME="E8:13_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="kurato_2.html#E7:13_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="kurato_2.html#D2" target="_self">Def2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:13"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:13"/><i><font color="Green">E11</font></i>: 
(  <a href="kurato_2.html#K5" target="_self">lim_sup</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K2" target="_self">meet</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K1" target="_self">Union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="kurato_2.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:13"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E1:13_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K1" target="_self">Union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E4:13"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:13_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="kurato_2.html#E2:13"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E4:13_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="kurato_2.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E5:13_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E1:13_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="kurato_2.html#E3:13_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:13"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K5" target="_self">lim_sup</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E4:13"><i><font color="Green">E11</font></i></a>, <a class="ref" href="kurato_2.html#T6" target="_self">Th6</a></i>;<br/>


</div>
